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A Theoretical Distribution for Mental Test Scores

Published online by Cambridge University Press:  01 January 2025

J. A. Keats  and
Frederic M. Lord
J. A. Keats
Affiliation:
University of Queensland, Australia
Frederic M. Lord
Affiliation:
Educational Testing Service*
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Abstract

The negative hypergeometric distribution of raw scores on mental tests is derived from certain assumptions relating to test theory. This result is checked empirically in a number of examples. Further derivations lead to the bivariate distribution of parallel tests which is also verified with actual data. The bivariate distribution of raw score and true score is also derived from a further assumption. This distribution is used to set confidence limits for true scores for persons with a given raw score.

Type
Original Paper
Information
Psychometrika , Volume 27 , Issue 1 , March 1962 , pp. 59 - 72
Copyright
Copyright © 1962 The Psychometric Society

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Footnotes

*

This work was supported in part by contract Nonr-2752(00) between the Office of Naval Research and Educational Testing Service. Reproduction in whole or in part for any purpose of the United States Government is permitted.

*

This approximation is believed to be a good one for the present purpose of finding practical mathematical forms for representing the distribution of x and the distribution of p. The fact that this approximation leads in several eases to the Kuder-Richardson formula-21 reliability coeffiemnt, as will be shown in due course, is not to be interpreted as suggesting the use of this formula in place of formula 20, however. Such a choice of reliability formulas should be based on test reliability theory, which is a much more rally developed theory than any presently available for representing g(x) and f(p).

References

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